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Full text of "Analytical Mechanics"

112
ANALYTICAL MECHANICS

to be fitted on a smooth horizontal rod (Fig. 62) so that they
are free to slide along it.    If the rod is rotated about a verti-
cal axis the bodies fly away from the axis of rotation.    If;
however, the bodies are connected by a string of negligible
mass they occupy positions
on the two sides of the axis,
which depend upon the ratio
of the masses.   So far as the
motion along the rod is con-
cerned, each body is equiv-
alent to a particle of the
same mass placed at  the
center of mass of the body.*
Suppose, as it is assumed in Fig. 62, the horizontal rod to be hollow and to have                    FIG. 62.
smooth inner wall; further
suppose the centers of mass of the given bodies to lie on the axis of the rod. Then if at the center of mass of each body a particle of equal mass is placed and the two particles connected by means of a massless string of proper lengths, the positions of the particles will remain at the centers of mass of the given bodies even when the rod is set rotating about the vertical axis.
Now let mi and w2 be the masses of the particles and fi and /2 their accelerations due to the rotation of the tube about the vertical axis. Then since the tensile force in the string is the same at its two ends, the forces acting upon the particles are equal. Therefore we have
F =
= w2/2,
or
/I
* For a proof of this statement see p. 242.